Theorem cmcm4i

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmcm4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	1 2	cmcm2i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) )
4	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
5	1 4	cmcm3i	⊢ ( 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) )
6	3 5	bitri	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ ( ⊥ ‘ 𝐵 ) )